# Properties

 Label 7728.2.a.bu Level $7728$ Weight $2$ Character orbit 7728.a Self dual yes Analytic conductor $61.708$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7728.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$61.7083906820$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.621.1 Defining polynomial: $$x^{3} - 6 x - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3864) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} + ( -1 - \beta_{1} + \beta_{2} ) q^{5} - q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + ( -1 - \beta_{1} + \beta_{2} ) q^{5} - q^{7} + q^{9} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{11} + ( -3 - \beta_{1} - 2 \beta_{2} ) q^{13} + ( -1 - \beta_{1} + \beta_{2} ) q^{15} + \beta_{2} q^{17} + ( -2 + \beta_{2} ) q^{19} - q^{21} - q^{23} + ( 8 - \beta_{1} ) q^{25} + q^{27} + ( -2 - 3 \beta_{2} ) q^{29} + ( -2 \beta_{1} + \beta_{2} ) q^{31} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{33} + ( 1 + \beta_{1} - \beta_{2} ) q^{35} + ( 2 \beta_{1} - \beta_{2} ) q^{37} + ( -3 - \beta_{1} - 2 \beta_{2} ) q^{39} + ( 4 - 2 \beta_{1} + \beta_{2} ) q^{41} + ( 3 - 3 \beta_{1} ) q^{43} + ( -1 - \beta_{1} + \beta_{2} ) q^{45} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{47} + q^{49} + \beta_{2} q^{51} + ( -3 + \beta_{1} + \beta_{2} ) q^{53} + ( 1 + 3 \beta_{1} + 7 \beta_{2} ) q^{55} + ( -2 + \beta_{2} ) q^{57} + ( 9 - \beta_{1} - \beta_{2} ) q^{59} + ( -11 + \beta_{1} ) q^{61} - q^{63} + ( -6 + 10 \beta_{1} + \beta_{2} ) q^{65} + ( -5 + \beta_{1} + \beta_{2} ) q^{67} - q^{69} + ( -1 + 5 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 6 - 4 \beta_{1} - \beta_{2} ) q^{73} + ( 8 - \beta_{1} ) q^{75} + ( -2 + 2 \beta_{1} + \beta_{2} ) q^{77} + ( -4 \beta_{1} - \beta_{2} ) q^{79} + q^{81} + ( 4 - 2 \beta_{1} - \beta_{2} ) q^{83} + ( 7 - 3 \beta_{1} - \beta_{2} ) q^{85} + ( -2 - 3 \beta_{2} ) q^{87} + ( 1 - \beta_{1} + 4 \beta_{2} ) q^{89} + ( 3 + \beta_{1} + 2 \beta_{2} ) q^{91} + ( -2 \beta_{1} + \beta_{2} ) q^{93} + ( 9 - \beta_{1} - 3 \beta_{2} ) q^{95} + ( 2 + 2 \beta_{1} + 5 \beta_{2} ) q^{97} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{3} - 3q^{5} - 3q^{7} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{3} - 3q^{5} - 3q^{7} + 3q^{9} + 6q^{11} - 9q^{13} - 3q^{15} - 6q^{19} - 3q^{21} - 3q^{23} + 24q^{25} + 3q^{27} - 6q^{29} + 6q^{33} + 3q^{35} - 9q^{39} + 12q^{41} + 9q^{43} - 3q^{45} + 3q^{49} - 9q^{53} + 3q^{55} - 6q^{57} + 27q^{59} - 33q^{61} - 3q^{63} - 18q^{65} - 15q^{67} - 3q^{69} - 3q^{71} + 18q^{73} + 24q^{75} - 6q^{77} + 3q^{81} + 12q^{83} + 21q^{85} - 6q^{87} + 3q^{89} + 9q^{91} + 27q^{95} + 6q^{97} + 6q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 6 x - 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 4$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.523976 2.66908 −2.14510
0 1.00000 0 −3.67750 0 −1.00000 0 1.00000 0
1.2 0 1.00000 0 −3.21417 0 −1.00000 0 1.00000 0
1.3 0 1.00000 0 3.89167 0 −1.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$1$$
$$23$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7728.2.a.bu 3
4.b odd 2 1 3864.2.a.k 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3864.2.a.k 3 4.b odd 2 1
7728.2.a.bu 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7728))$$:

 $$T_{5}^{3} + 3 T_{5}^{2} - 15 T_{5} - 46$$ $$T_{11}^{3} - 6 T_{11}^{2} - 15 T_{11} + 84$$ $$T_{13}^{3} + 9 T_{13}^{2} - 9 T_{13} - 164$$ $$T_{17}^{3} - 9 T_{17} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$-46 - 15 T + 3 T^{2} + T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$84 - 15 T - 6 T^{2} + T^{3}$$
$13$ $$-164 - 9 T + 9 T^{2} + T^{3}$$
$17$ $$4 - 9 T + T^{3}$$
$19$ $$-6 + 3 T + 6 T^{2} + T^{3}$$
$23$ $$( 1 + T )^{3}$$
$29$ $$-262 - 69 T + 6 T^{2} + T^{3}$$
$31$ $$-74 - 39 T + T^{3}$$
$37$ $$74 - 39 T + T^{3}$$
$41$ $$18 + 9 T - 12 T^{2} + T^{3}$$
$43$ $$216 - 27 T - 9 T^{2} + T^{3}$$
$47$ $$-232 - 72 T + T^{3}$$
$53$ $$-2 + 15 T + 9 T^{2} + T^{3}$$
$59$ $$-628 + 231 T - 27 T^{2} + T^{3}$$
$61$ $$1262 + 357 T + 33 T^{2} + T^{3}$$
$67$ $$72 + 63 T + 15 T^{2} + T^{3}$$
$71$ $$-828 - 153 T + 3 T^{2} + T^{3}$$
$73$ $$686 + 15 T - 18 T^{2} + T^{3}$$
$79$ $$344 - 93 T + T^{3}$$
$83$ $$82 + 21 T - 12 T^{2} + T^{3}$$
$89$ $$24 - 159 T - 3 T^{2} + T^{3}$$
$97$ $$1656 - 207 T - 6 T^{2} + T^{3}$$